On a Diophantine Inequality with Primes Yielding Square-Free Sums with Given Numbers

TL;DR

The paper proves that for irrational $\alpha$ and real $\beta$, and for distinct positive integers $a_1<\dots<a_s$ that do not form a full reduced residue system modulo $p^2$ for any prime $p$, there exist infinitely many primes $p$ with $||\alpha p+\beta||<p^{-\theta}$ for any $\theta<\tfrac{1}{10}$, while all numbers $p+a_1,\dots,p+a_s$ are square-free. The authors implement a smoothing via a 1-periodic function $\chi$ and analyze a weighted prime sum $\Gamma(x)$, decomposing it into a main term $\Delta\Gamma_1(x)$ and a secondary term $\Delta\Gamma_2(x)$. The main term is shown to contribute a positive singular-series-like term using Changa-type asymptotics, while the secondary term $\Gamma_2$ is bounded sharply through a sequence of exponential-sum estimates, Heath-Brown’s identity, and Heath–Vaughan–Matomäki-type bounds. This combination yields a lower bound on $\Gamma(x)$ for infinitely many $x$, proving the claimed infinitude result. The work advances Diophantine approximation on primes under arithmetic restrictions by exploiting intricate decompositions of divisor-type sums and refined exponential-sum techniques.

Abstract

Let $α\in \mathbb{R}\setminus\mathbb{Q}$ and $β\in \mathbb{R}$ be given. Suppose that $a_1,\ldots,a_s$ are distinct positive integers that do not contain a reduced residue system modulo $p^2$ for any prime $p$. We prove that there exist infinitely many primes $p$ such that the inequality $||αp+β||<p^{-1/10}$ holds and all the numbers $p+a_1,\ldots,p+a_s$ are square-free.

Theorems & Definitions (10)

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